Day 10  Equations of Circles, Circumference, and Area

You can write an equation of a circle in a coordinate plane if you know its radius and the coordinates of its center.  Suppose the radius of a circle is r and the center is (h, k).  Let (x, y) be any point on the circle.  The distance between (x, y) and (h, k) is r, so you can use the Distance Formula.

If we square both sides, we get the standard equation for our circle with radius r and center (h, k).

Standard Equation of a Circle: (x  h)2 + (y  k)2 = r2

If the center is the origin, then the standard equation is x2 + y2 = r2.

GRAPHING CIRCLES

If you know the equation of a circle, you can graph the circle by identifying its center and radius.  Heres an example.  We have an equation for a circle being (x + 2)2 + (y  3)2 = 9.  We can rewrite this to find our center and radius.

(x  (-2)2) + (y  3)2 = 32: So our center is (-2, 3) and radius = 3.  Using this, our graph is:

Circumference of Circles

The circumference of a circle is the distance around the circle (just as the perimeter of a polygon is the distance around the polygon).  The ratio of the circumference C of a circle to its diameter d is a number slightly greater than 3, regardless of the size of the circle.  The ratio, by definition, of the circumference of a circle to its diameter is the irrational number called pi or .  We have this theorem:

If a circle has a circumference of C units and a radius of r units, then C = 2r or C=d

The exact circumference would be a measurement of pi.  When asked to find the circumference, usually you are looking for an estimate of the circumference.

Area of a Circle

When we want to measure the amount of space occupied by a circle, we are looking for the area.  As the number of sides of a polygon inscribed in a circle increases, the area of the polygon approached the value of r2.

If a circle has an area of A square units and a radius of r units, then A =r2

A sector of a circle is the region bounded by a central angle (or two radii) and its corresponding arc.  If we want to find just the area of a sector of the circle, we have a special formula.

If a sector of a circle has an area of A square units, a central angle measurement of N degrees, and a radius of r units, then .